Local energy minimizers for elastic bars with cohesive energy

At the level of constitutive equations relating macroscopic stress and strain 
measures, real materials exhibit a variety of behaviors. In the years, there 
was a tendency to focus on some prominent properties of material response and 
to study them individually; the result was the development of independent, 
sometimes unrelated, branches of continuum mechanics, such as fracture 
mechanics, damage, and many theories sharing the name of plasticity. 
Some hope for recovering a unified view is provided by a model based 
on the assumption that the total energy of a body is the sum of two parts, 
a bulk part representing the elastic strain energy and a cohesive part 
associated with defects occurring at both macroscopic and microscopic level. 
Energy minimization provides a number of stable and metastable equilibrium 
curves, whose shape strongly depends on the analytic expression assumed for 
the surface energy. Different expressions reproduce different types of material
behavior. The wide range covered by the model may contribute in establishing 
a link between macroscopic response of bodies of finite size and microscopic 
properties of the constituent materials.
At the present stage, the model has been applied to one-dimensional, 
homogeneous bars with prescribed displacements at the ends. Non-homogeneous 
bars subject to applied body forces have been considered recently. Two and 
three-dimensional generalizations present considerable difficulties, both in 
mechanical modeling and in mathematical setting. The present communication is 
an updated review of the state of the research, with emphasis on some peculiar 
aspects of the mathematical structure of the model.